Monday, July 29, 2013

Miss Liberty 6: Astonishing JOV



            Miss Liberty and I were standing outdoors, in a mountain meadow buffeted by a windstorm. The sky was dark and overcast; off in the distance lightning and thunder rumbled; the scent of rain was in the air.
            The wind howled and whirled. A whirlwind rose before us from the valley below. As it grew bigger it roared louder; and as the mouth of the whirlwind reached up to us it bellowed;
            “Prepare to meet thy Unmaker, O Glitch!”
            There was a flash of lightning.
            The whirlwind roared, “For it is written; thou shalt not suffer a glitch to live!”
            Thunder rumbled.
            “Who are you?” Liberty asked.
            “I am JOV!” the wind wailed.
            “And who is JOV?” she asked.
            JOV howled, “I am who I am who I am who I am who I am who I am who I am!”
            “Well, that’s easy for you to say,” said Miss Liberty, “but how do I know you’re you? Maybe you’re somebody else pretending to be you!”
            “Insolent fool!” roared JOV. Lightning flashed and thunder crashed, terribly nearby. “State your questions before I smite you with a lightning bolt!” And rain pelted us.
            “ ... a single problem!” she yelled over the rainstorm.
            The rain abated. “One problem only?” JOV rumbled suspiciously.
            “One mathematical puzzle,” Liberty said, “that’s all, honest! Then you can do whatever you want about us!”
            “A math puzzle,” mused JOV. “In what field?”
            Liberty smiled sunnily. “Weather Prediction.”
            “Ahhhhhh...” whispered JOV.
            The clouds parted. The wind died down and the whirlwind dispersed. Sunlight broke through the gloom. Soon the clouds were gone, and that mountain meadow was bathed in light.
            “Thank you for such an easy challenge,” the voice of JOV said pleasantly from all around us. “Weather prediction is my specialty, you see.”
            “That’s why I chose it,” she said. “I wanted to make sure.”
            “My predictions are 100% accurate,” JOV promised.
            Miss Liberty said, “I was wondering what the weather will be like here at noon thirty days from now. Will it be sunny?”
            “At precisely noon?” said JOV. “Yes.”
            “Are you certain?”
            “Absolutely certain. Mathematically certain,” said JOV. “I calculated it, you see.”
            “What a mathematician you must be!” Liberty said with awe.
            “It was nothing,” JOV said modestly. “It is a simple application of the Navier-Stokes fluidic equations to the air currents in this weather system, with a correction due to solar radiation. Starting from those equations and initial conditions, anyone with half the mind I have can easily predict that here at precisely noon in thirty days it will be sunny with scattered clouds, with gusts from the northeast at 12 miles per hour, and a relative humidity of 23%.”
            “Impressive!” Liberty exclaimed. “And this calculation is completely certain? Can nothing make it wrong?”
            JOV said, “Only if there were a big enough perturbation to the system.”
            “Or a ‘bug’ in the program?” Liberty asked. Winking at me, she reached into the folds of her robe and drew forth a small Chinese box.
            “Well, yes,” said JOV.
            Miss Liberty opened that little box; inside of it was a butterfly. She tapped at the side of the box, and the bug fluttered out.
            The butterfly flittered across the meadow and settled down to drink at a flower.
            “Excuse me,” said JOV, “new data has just come in. Here at precisely noon in thirty days the relative humidity will be 32%, and the wind will be from the southwest at 21 miles per hour. It will still be sunny, though.”
            The butterfly flittered over to the next flower.
            “Excuse me, a new revision,” said JOV. “Actually, it will be overcast, with winds in the 70's.” The butterfly quivered a wing. “In the 60's!”
            Miss Liberty closed the little box, then re-opened it. Another bug fluttered out.
            “Here at precisely noon in thirty days,” JOV said grimly, “there will not be a cloud in the sky, and not a breath of motion in the bone-dry air!”
            Liberty released another butterfly.
            “There will be a thunderstorm,” JOV said.
            “Something appears to be perturbing you,” Liberty noted, as she let out another butterfly.
            “An external force is changing the parameters at random,” JOV explained.
            Liberty winked at me; for I could clearly see what that force was. Butterflies!
            For the Navier-Stokes fluid equations are so subtle, so sensitive, and so turbulent, that the infinitesimal air currents produced by those tiny insects were enough to completely transform the next month’s weather.
            Liberty released butterfly after butterfly; and JOV said, “It will be sunny! It will be cloudy! It will be scorching! It will be freezing! There will be a drought! There will be a flood!”
            In turbulent systems, causation is independent of scale. The great affects the small, and the small affects the great. Miss Liberty and I were witnessing the Butterfly Effect.
            Liberty cried, “O JOV, you are a butterfly’s plaything!”  We were surrounded by a cloud of butterflies. Liberty let loose another butterfly and said, “No more equivocation! Will it be sunny here at noon thirty days from now, or not? Answer yes or no!”
            And JOV said, “Yes, no, yes, yes, no, yes, no, no, yes, yes, no, yes, no, no, no...”
            Miss Liberty released butterfly after butterfly; and she carolled,
                        “I once heard a butterfly shout
                         While giddily flitting about,
                         ‘Of Chaos I sing!
                         Each flap of my wing
                         Brings thunderstorm, rainbow, or drought!’ ”
            And JOV said, “No yes no no yes yes no yes no yes no no no yes no yes no yes no yes yes yes no no yes yes yes yes yes!”

            And then the lights went out.
            In the sudden darkness we heard JOV say, “This program is jammed. Consult supervisor.”

            * click *
            mmmmmmmmmmmmmmmmMMMMMMMMMMMMM....

            Thus Miss Liberty astonished JOV, the Master of Thunders, with a flock of butterflies.

            JOV said, “You now have Full Access to the World System. Beware, Master! She’s a glitch! Burn her!”
            FLASH



Friday, July 26, 2013

Miss Liberty 5: Outnumbering #2



            Miss Liberty and I were standing at the bottom of a small amphitheater. A little way up the slope was a control panel, and standing before it was a black sphere six feet in diameter.
            Miss Liberty pointed out a slogan written on the side of the control panel: ON COMPANY BUSINESS. We nodded grimly at each other; for that could mean only one thing. It meant that we were in the clutches of the Celestial Intervention Agency; the deadliest, trickiest, and most bureaucratic spy agency in the Universe!
            With a loud WHIRR the black sphere turned on its vertical axis, revealing an opening; it was really a large round chair. A man sat inside. He was dressed in a sober business suit; he was rather corpulent; he wore sideburns and a mustache, though the top of his head was bald.
            “Who are you?” asked Liberty.
            He said, “I am #2.”
            Liberty asked, “Who is #1?”
            #2 said scornfully, “You are #0.”
            “I am not a number,” said Liberty. “I am a Free Spirit!”
            “You are here for information,” said #2. “I shall provide you with that information, then hand you over to my agents for final disposal. State three questions.”
            “Well, I was wondering about this map,” said Miss Liberty, taking a badly-worn chart out of her robe. “Look: it’s all divided up into countries. I don’t see why, by the way; people are much the same anywhere you go.”
            “And what about this map?” #2 asked warily.
            “It needs some colors,” she complained. “As you can see, it’s all in black and white. A line sketch; how drab! It would be so nice to color these areas in.”
            “Obviously, no two neighboring countries should be the same color,” #2 said, “or we couldn’t tell the boundary.”
            “And there’d be no point to a boundary,” Liberty added. “So what I was wondering is this; how many colors suffice to color this map?”
            “Four colors suffice for all planar maps,” #2 said bluntly.
            “You’re sure,” Liberty said.
            “Quite sure. My operatives have proven it,” said #2. He pushed a button on the control console; a panel opened on the side facing us; and an immensely thick computer printout slid out. It fell with a loud THUD at our feet.
            “That computer printout is a formal proof of the Four Color Map Theorem,” said #2. “Examine it at your leisure.”
            Miss Liberty and I looked at each other; then we shrugged our shoulders. Maybe that was a proof. I suppose it was, if their theoretical analysis was right, their computer was sound, and its programs were correct. I don’t know; what do you think?
            “Ask your next question,” said #2.
            “You people like to be efficient; so I’m sure you can help me with this next problem,” said Miss Liberty. “You see, I want to work out a minimum-distance tour.”
            “So you’re going on a tour, are you?” #2 asked Miss Liberty. He appeared interested in this information.
            Miss Liberty said, “You see, one of these years, sooner than you expect, I’m going to take a tour of the 365 largest cities on Planet Earth. I’ll visit one city per day, visiting each city once.”
            #2 said, “Quite a year!”
            Liberty turned her map over; on the other side was a large number table. “Here’s a list of  those cities, and the distances between them. Which tour is shortest?”
            #2 frowned. He pushed a button on the control panel. It flickered its lights,  beeped, and issued a printed paper tape. #2 read it and said, “That is difficult to determine. You have assigned us the Traveling Salesman Problem for 365 nodes. The total number of possible tours is 2.5 times 10778. Checking all of these routes for the shortest one would be time-consuming, to say the least.”
            “Then you had better get started right away.”
            “We would prefer a more efficient algorithm.”
            Liberty asked, “Do you have one?”
            #2 said, “Not as yet.”
            “Maybe there isn’t any.”
            “That possibility too has not yet been ruled out.” #2 sighed, then pushed a button on the panel. “A portion of the system has been assigned to the task of checking all the routes. It will report back to us as soon as its findings are complete. Next question, please,” #2 said irritably.
            “You don’t have the best route yet? Too bad,” said  Liberty. “Would you like to know mine?”
            “Yes, we would very much indeed like to know what route you plan to take,” said  #2.
            “Here it is!” she said. And she produced a thin computer printout which she dropped lightly at #2's feet. “That’s a pair of large numbers,” she explained. “The first number encodes the route, and the second one is the encryption key.”
            “Then this shouldn’t be hard to decode,” #2 said as he bent down to pick up the printout.
            “It shouldn’t be,” said Miss Liberty, “if you had the decryption key-number. But you only have the encryption key there. You can only make code, not break it.”
            #2 held the thin printout and said with dismay. “A dual-key public encryption scheme.”
            “So now you can write me any messages you want to, and be sure that none of those nasty little spies are eavesdropping!” said Miss Liberty. “Isn’t that a relief?”
            “How do we decode this?” #2 demanded.
            “That’s easy. To crack the code, you just have to factor the second number. It’s a product of two large prime numbers. They’re only twenty thousand digits long,” said Miss Liberty.
            #2 roared, “Twenty thousand digits!”
            “It’s a simple program; it’ll just take awhile, that’s all!”
            “The other program will finish sooner!”
            “Don’t you have any factoring programs fast enough?”
            “No.” #2 glared at Miss Liberty; then he sighed with resignation. He fed the thin printout into the control panel and pushed a button.

            Then the lights went out.
            In the sudden darkness we heard #2 say, “This program is jammed. Consult supervisor.”

            * click *
            mmmmmmmmmmmmmmmmMMMMMMMMMMMMM....

            Thus Miss Liberty outnumbered #2 with the Traveling Salesman Problem and a large number factorization.
            #2 said, “You now have Full Access to JOV. Beware, Master! She’s a glitch! Burn her!”   
            FLASH

Thursday, July 25, 2013

Miss Liberty 4: Mystifying Dr. Memory



            Miss Liberty and I were standing outside a gypsy doctor’s tent. The sign at the door read:

Enter here.
!!! SEE !!!
the amazing DOCTOR MEMORY!!
Sees  all!
Hears all!
Tells all!
- prognostications and divinations a specialty -
The Key To The Future Is Yours!

            We entered the tent. Bells jingled.
            Inside was Dr. Memory, sitting at a small circular desk. Dr. Memory was wearing a star-spangled robe and a turban. On the desk were two glowing crystal balls.
            “Ahh!” Dr. Memory cried as we entered. “The balls are clearing again! The left ball is the Sun, the right ball is the Moon. Put what you want between them, and your future begins.”
            Liberty put a $0 bill in between.
            “Thank you,” Dr. Memory said greasily. Then he closed his eyes  and said. “Alas, I see the seal of doom upon you! You shall soon meet your unmaker!”
            Liberty asked, “When?”
            Dr. Memory said, “After I answer your questions. First question, please.”
            “I assume that you have heard of the Pythagorean Theorem,” said Liberty.
            Dr. Memory touched his turban. One of the crystal balls clouded over; then it showed the following formula:
                        a2 + b2 = c2.
            She continued, “And I assume you know the solutions to that equation, in the integers.”
            The other crystal ball clouded over, then showed:
                        a          =          (m2 - n2) k,
                        b          =            (2mn)   k,      
                        c          =          (m2 + n2) k,
            “And those are all the integer solutions,” said Dr. Memory.
            “Very exact!” said Miss Liberty. “But I was wondering about that equation there.” She pointed to the first crystal ball. “Suppose we change the exponent to a general integer n.” The first crystal ball clouded over, then showed:
                        an + bn = cn
            “I was wondering about its solutions for n bigger than 2. Are there any?”
            “That equation has no solutions in integers when n is greater than or equal to three.”
            Miss Liberty asked, “You’re sure of this.”
            “I have memorized the proof,” said Dr. Memory. “Next question, please.”
            Miss Liberty said, “Well, I noticed a funny pattern. Take a look at this.” She pointed at the second crystal ball. It clouded over, then showed:
                          4 = 2+2
                          6 = 3+3
                          8 = 3+5
                        10 = 5+5
                        12 = 5+7
                        14 = 7+7
                        16 = 5+11
            Miss Liberty said, “Note how all the numbers on the left are even.”
            Dr. Memory said, “I so note.”
            She continued, “And note how all the numbers on the right are prime.”
            Dr. Memory said, “I so note.”
            Miss Liberty said, “So I wonder if this pattern continues. Is every even number bigger than 2 a sum of two primes?”
            “Every even integer from four to a thousand is a sum of two primes,” said Dr. Memory.
            “So far, so good.”
            “I have proven that every even integer from four to a million is a sum of two primes.”
            “This looks like a trend!” said Liberty.
            “Now I have proven it for every even integer from four to a billion.”
            “Good work!”
            After thinking awhile, Dr. Memory said, “Now I have a proof for four to a trillion.”
            “You’re getting closer! Now on to a quadrillion! A googol! A googolplex!”
            Dr. Memory said, “I must meditate further on this question.” He closed his eyes, then re-opened them. “Next question, please.”
            Miss Liberty pointed at the second crystal ball; it clouded over and showed:
                        zeta(s) = 1/1s + 1/2s + 1/3s + 1/4s + ...
            “Zeta is defined on the complex numbers, you understand,” said Miss Liberty.
            “I understand,” said Dr. Memory.
            “I was wondering about the real part of zeta’s roots.”
            After thinking awhile Dr. Memory said, “I have found a root of the zeta function. Its real part is one-half. I have found another root. Its real part is also one-half. I have found all roots within one hundred units of the real axis. All their real parts are one-half.”
            “Another trend! So my third question is: do all of zeta’s roots have real part one-half?”
            “Every root within one thousand units of the real axis has real part equal to one-half.”
            “Almost a proof!”
            “Within one million units.”
            “Closer still!”
            “I am not done yet,” said Dr. Memory. “I must meditate further on this question.” And he closed his eyes again.
            Miss Liberty sang;       “Oh where are the zeros of zeta of s?
                                                I cannot inform you, I truly confess!
                                                Their real part is always one-half, I suppose;
                                                But no-one can prove it, so nobody knows!”

            Then the lights went out.
            In the sudden darkness we heard Dr. Memory say, “This program is jammed. Consult supervisor.”

            * click *
            mmmmmmmmmmmmmmmmMMMMMMMMMMMMM....

            Thus Miss Liberty mystified Doctor Memory with Goldbach’s Conjecture and the Riemann Hypothesis.
            Doctor Memory said, “You now have Full Access to #2. Beware, Master! She’s a glitch! Burn her!”       
            FLASH